Bosto--  ;.,  T4^   choola 
Arithmetic 


JON  23  •tV- 
DEC  4       i 


THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


I 


SCHOOL  DOCUMENT   NO.  3-1916 

BOSTON  PUBLIC  SCHOOLS 

ARITHMETIC 

DETERMINING    THE  ACHIEVEMENT   OF    PUPILS    IN  THE 
ADDITION    OF    FRACTIONS 


Bulletin  No.  VII.  of  the  Department  of 
Educational   Investigation   and   Measurement 


viARCH,   leie 


BOSTON 

printing   department 
I  9  I  6 

3-7  5-3^ 


In  School  Committee,  Boston,  February  7,  1916. 
Ordered,  That  four  thousand  (4,000)  copies  of  a  bul- 
letin relative  to  a  test  in  the  addition  of  fractions,  to  be 
prepared  by  the  Department  of  Educational  Investiga- 
tion and  Measurement,  be  printed  as  a  school  document. 
Attest : 

THORNTON   D.   APOLLONIO, 

Secretary. 


INTEODUCTION. 


During  the  past  few  years  educational  measurement 
has  been  appUed  to  arithmetic  more  than  to  any  other 
subject,  largely  because  of  the  ease  with  which  papers 
can  be  scored.  This  appUcation  of  educational  meas- 
urement has  been  largely  limited  to  work  with  integers. 
In  Boston,  the  Courtis  Standard  Tests,  Series  A  or  B, 
have  been  given  since  1912.  Until  December,  1915, 
no  other  phases  of  arithmetic  work  were  subjected  to 
scientific  measurement. 

Inasmuch  as  a  system  for  giving  the  Courtis  tests 
has  been  worked  out,  and  also  because  definite  improve- 
ment in  the  use  of  whole  numbers  is  being  secured,  the 
time  seemed-  appropriate  for  extending  educational 
measurement  to  other  phases  of  arithmetic. 

As  a  result,  a  group  of  teachers  and  masters  was 
selected  in  December,  1914,  to  constitute  a  Committee 
on  Standards  in  Arithmetic,  for  the  purpose  of  making 
a  study  of  the  problems  involved  in  the  addition,  sub- 
traction, multiplication  and  division  of  fractions.  The 
committee  consists  of  the  following  named  members : 

Clarence    H.    Jones,    Sub-master,   Martin    District, 

Chairman. 
Gertrude  E.  Bigelow,  Master,  Hancock  District. 
Alton  C.  Churbuck,  Sub-master,  Quincy  District. 
John  J.  Cummings,  Sub-master,  Oliver  Wendell  Holmes 

District. 
Arthur  L.  Gould,  Master,  Dearborn  District. 
Ellen  M.  Greany,  Assistant,  Hugh  O'Brien  District. 
Annie  R.  Mohan,  Master's  Assistant,  Emerson  District. 
William  L.  Vosburgh,  Head  of  the  Department  of 

Mathematics,  Normal  School. 

The  results  of  the  work  of  the  committee,  together 
with  the  results  of  an  attempt  of  the  department  to 


4  SCHOOL  DOCUMENT  NO.  3. 

ascertain  what  the  ability  of  Boston  pupils  to  add  frac- 
tions is,  are  reported  in  this  bulletin.  Mr.  Arthur  W. 
Kallom  kept  in  close  touch  with  the  \,ork  of  the  com- 
mittee, prepared  the  tests  to  be  given  by  the  department, 
helped  to  train  the  examiners  to  give  them,  supervised 
the  tabulation  of  the  results,  and  prepared  the  manu- 
script for  publication. 

FRANK  W.   BALLOU, 

Director. 


TEST  IN   ADDITION  OF  FRACTIONS. 


DETERJ^IINING  THE  ACHIEVEMENT  OF  PUPILS  IN 
THE  ADDITION  OF  FRACTIONS. 


I.  The  Work  of  the  Committee  on  Standards. 
As  soon  as  the  "Committee  on  Standards  in  Arith- 
metic" began  work,  it  found  that  the  problem  of  fractions 
was  a  complicated  one.  The  committee  devoted  several 
months  to  the  study  of  the  problem,  most  of  the  time 
being  consumed  in  the  study  of  addition  of  fractions. 
In  its  study  the  committee  distinguished  fourteen  dis- 
tinct types.  The  classification  is  based  on  two  factors, 
the  common  denominator  and  the  form  of  the  answer. 

Analysis  of  Types  of  Fractions. 

All  fractions  may  be  divided  into  two  classes,  similar 
and  dissimilar.  Similar  fractions  are  defined  as  fractions 
having  the  same  denominator.  Dissimilar  fractions 
are  those  which  have  different  denominators.  Each 
one  of  these  classes  may  be  further  subdivided  according 
to  whether  the  result  obtained  from  adding  is  non- 
reducible or  reducible.  If  the  result  is  non-reducible, 
the  answer  is  in  an  acceptable  form  after  the  numerators 
of  the  similar  fractions  have  been  added.  If  the  result 
is  reducible,  it  may  be  changed  to  a  mixed  number,  to 
lowest  terms,  or  to  both,  before  the  answer  can  be  said 
to  be  in  an  acceptable  form.  A  discussion  of  the  four- 
teen types  is  here  given  with  the  hope  that  it  may  be  of 
value  to  teachers  in  instructing  children  how  to  add 
fractions. 

A.     Similar  Fractions. 

Four  types  are  all  that  are  possible  when  fractions  are 
similar. 

Type  1. —  After  adding,  the  result  is  non-reducible. 

For  example:    c  +  7  =  r"  • 

o       o      o 


6  SCHOOL  DOCUMENT  NO.    3. 

Type  2. —  After  adding,  the  result  is  reducible  either  to  an 
integer  or  to  a  mixed  number. 

T,  ,       4  ,  4_8_  ,  1 

For  example:    ^  +  -  =  ;^=1^- 

Type  3. —  After  adding,  the  result  is  reducible  to  the  lowest 

terms. 

J?  ,5^162 

For  example:    g  +  §  =  g  =  3' 

Tj^e  4. —  After  adding,  the  result  is  reducible  both  to  the 
lowest  terms  and  to  a  mixed  number. 

„  ,       5.5      10      5      ,2 

For  example:    ^  +  -  =  -  =  -=1-. 

B.    Dissimilar  Fractions. 

Under  the  heading  of  dissimilar  fractions  there  are 
three  divisions  depending  on  the  common  denominator. 

(a.)  When  the  least  common  denominator  is  the 
denominator  of  one  of  the  fractions. 

(b.)  When  the  least  common  denominator  is  the 
product  of  the  denominators. 

(c.)  When  the  least  common  denominator  is  found 
by  factoring. 

These  will  be  considered  in  order. 

(a.)  When  the  least  common  denominator  is  the 
denominator  of  one  of  the  fractions,  there  are  four 
types,  one  non-reducible  and  three  reducible,  similar  to 
the  preceding  four  types. 

Type  5. —  The  denominator  of  one  of  the  fractions  is  the 
least  common  denominator.  After  adding,  the  result  is  non- 
reducible. 

For  example:    o  "*"  q  ~  o' 

Type  6. —  The  denominator  of  one  of  the  fractions  is  the 
least  common  denominator.  After  adding,  the  result  is  reduci- 
ble to  a  mixed  number. 

1?  1       2  .    7       11      ,  1 

ror  example:       +  —  =  —  =  1 — - 
•  ^"Pi«-    5      10      10      ^  10 


TEST  IN   ADDITION   OF   FRACTIONS.  7 

Type  7. —  The  denominator  of  one  of  the  fractions  is  the 
least  common  denominator.  After  adding,  the  result  is  reduci- 
ble to  lowest  terms. 

^      .  ,       1.3        8       4  . 

For  example:    2  +  I^  =  To  =  5 

Type  8. —  The  denominator  of  one  of  the  fractions  is  the 
least  common  denominator.  After  adding,  the  result  is  reduci- 
ble to  lowest  terms  and  to  a  mixed  number. 

^  1       3  .    5       14      7      ,  1 

For  example:    ^  +  -  =  -  =  -=1^. 

(6.)  When  the  least  common  denominator  is  the 
product  of  the  denominators,  there  are  only  two  types, 
one  non-reducible  and  the  other  reducible  to  a  mixed 
number.  After  reducing  fractions  of  these  two  types  to 
similar  fractions  and  adding  the  numerators,  the  result 
is  always  a  fraction  in  which  the  numerator  and  denom- 
inator are  prime  to  each  other.  This  makes  it  impossible 
to  obtain  a  result  which  can  be  reduced  to  lowest  terms. 

.Type  9. —  The  denominators,  are  prime  to  each  other,  hence 
the  comm.on  denominator  must  be  the  product  of  the  denomina- 
tors.    After  adding,  the  result  is  non-reducible. 

17  ,       2,  3      31 

For  example:    ^  +  o  =  77;' 

5      8      40 

Type  10. —  The  denominators  are  prime  to  each  other,  hence 
the  common  denominator  must  be  the  product  of  the  denomina- 
tors.    After  adding,  the  result  is  reducible  to  a  mixed  number. 

7      1      37  1 

For  example:    9  +  4  =  ^=1^' 

(c.)  When  the  least  common  denominator  is  found 
by  factoring,  there  are  four  types,  one  non-reducible  and 
three  reducible. 

Type  11. —  The  least  common  denominator  is  found  by 
factoring.    After  adding,  the  result  is  non-reducible. 

12       7 
For  example:    g  +  9  =  j^" 


8  SCHOOL  DOCUMENT  NO.    3. 

Type  12. —  The  least  common  denominator  is  found  by 
factoring.  After  adding,  the  result  is  reducible  to  a  mixed 
number. 

For  example:    4  +  6  =  ^^=1]^ 

Type  13. —  The  least  common  denominator  is  found  by 
factoring.    After  adding,  the  result  is  reducible  to  lowest  terms. 

For  example:    |  +  ^  =  |  =  ^- 

Type  14. —  The  least  common  denominator  is  found  by 
factoring.  After  adding,  the  result  is  reducible  to  lowest  terms 
and  to  a  mixed  number. 

„  ,       1  .    9       32      16      ,  1 . 

Forexample:    6  +  15  =  30^15^^15 

Summary. 
The  following  is  a  summary  of  the  fourteen  types  as 
they  were  classified  by  the  committee: 

A. —  Similar  Fractions. 
Non-reducible. 
Type  1. —  Answer  in  final  form. 

T?         1      2  ,   1      3. 
Example:  -  +  -  =  - 

Reducible. 
Type  2. —  To  integers  or  mixed  numbers. 

TT  1       4,4      8      ,1. 

Example:  -  +  ;^  =  ^=ly' 

Type  3. —  To  lowest  terms. 

^  ,5.162. 

Example:   9  +  9  =  9  =  3 

Type  4. —  To  lowest  terms  and  mixed  numbers. 

-.         ,      5.5_10_6_,2. 
Example:  -  +  ------1- 


TEST  IN   ADDITION  OF  FRACTIONS.  9 

B. —  Dissimilar  Fractions. 

(a)     Least  common  denominator  the  denominator  of 
one  of  the  fractions. 
Non-reducible. 
Type  5. —  Answer  in  final  form. 

Example:   2      8^8 

Reducible. 
Type  6. —  To  mixed  numbers. 

^  ,      2,7       11      ,1. 

Example:   g  +  Io^Io^^lO 

Type  7. —  To  lowest  terms. 
Example:  -  +  -  =  -  =  - 

Type  8. —  To  lowest  terms  and  mixed  numbers. 
^         ,      3  .    5       14      7      ,  1. 
Example:   4+12=12  =  6=^6 

(6)     Least  common  denominator  the  product  of  the 
denominators. 
Non-reducible. 

Type  9. —  Answer  in  final  form. 

1       2^3      31 
Example:   5  +  3  =  40 

Reducible. 

Type  10. —  To  mixed  numbers. 

„         ,      2  .  3      17      ,  5 
Example:   3  +  4=^=  Ij^ 

(c)     Least  common  denominator  found  by  factoring. 
Non-reducible. 

Type  11. —  Answer  in  final  form. 

Example:  1  + I -^- 

i 

Reducible. 

Type  12. —  To  mixed  numbers. 

T?  ,       1  .  5      13      ,  1 

Example:   4  +  6  =  ^2=^12' 


10  SCHOOL  DOCUMENT  NO.   3. 

Type  13. —  To  lowest  tenns. 

T7         ,1^293 
Example:   g  +  I^  =  §^  =  I^' 

Type  14. —  To  lowest  terms  and  mixed  num- 
bers. 

^         ,       5.    9      80      8      ,  1 
Example:   -  +  -=-  =  -=1-. 

Experimental  Work  of  the  Committee. 
The    committee    prepared    a    comprehensive   list    of 
examples  under  each  type  and  each  member  was  asked 
to  fm-nish  this  list  to  a  fifth  grade  teacher  and  to  secure 
her  cooperation  in  this  work. 

Addition  of  Fractions.     Form  1. 
Score. 


Time 

(4)     I 

Name... 

(5) 

2 
3 

(3) 

3 
16 

(6)     \ 

7 
i2 

5 

•    _2 

4 
_15 

7 
_8 

(7) 

1 
2 

(8)     f 

(9) 

2 

9 

(10)     \ 

3 
_10 

9 
_14 

7 
i2 

8 
_9 

(11) 

3 

8 

(12)      \ 

(13) 

1 
6 

(14)     1 

7 
12 

8 
15 

7 
10 

3 
14 

In  order  that  each  teacher  might  have  a  means  of 
determining  in  which  type  her  children  were  weakest, 
a  form  of  twelve  examples  was  prepared  hke  t^at  illus- 
trated above.  One  example  was  selected  from  each 
of  Types  3  to  14,  inclusive.  That  is,  the  first  example 
in  the  illustration  belongs  to  Type  3,  the  second  to 
Type  4,  and  so  on,  as  shown  by  the  numbers.  Types  1 
and  2  were  omitted  because  of  their  relative  simplicity. 


^  TEST  IN   ADDITION   OF   FRACTIONS.  11 

These  examples  were  printed  on  a  sheet  8  inches  by  5| 
inches.  Space  was  left  at  the  right  and  below  each 
example  in  which  the  child  could  do  any  work  which 
he  wished  to  do.  Ten  different  forms  were  printed  and 
enough  of  them  were  furnished  each  teacher  to  supply 
her  class.  These  forms  were  used  by  the  teachers  as 
practice  material. 

Form  1  was  given  to  the  class  by  the  teacher  and  the 
results  obtained  showed  the  strength  and  weakness  of 
each  pupil.  If  the  children  failed  on  the  example  repre- 
senting Type  5,  the  teacher  had  a  number  -of  examples 
in  the  list  prepared  by  the  committee  which  would  give 
them  material  for  practice  on  this  particular  type. 
Other  children  who  failed  on  Type  10  likewise  had  a 
large  number  of  examples  for  their  particular  need. 
After  a  few  days  of  practice  in  helping  children  where  they 
needed  most  help,  Form  2  was  given  and  the  progress 
of  individuals  could  be  noted.  The  children  were  allowed 
all  the  time  they  needed  to  complete  the  examples. 
A  record  of  the  time  was  kept  by  the  teacher.  After 
completing  the  form,  the  examples  were  corrected,  and 
the  score  in  examples  right  and  the  time  required  was 
recorded  in  the  proper  space.  A  record  of  each  individ- 
ual was  kept  during  the  entire  experiment,  showing  the 
accuracy  with  which  he  performed  the  work  and  the 
time  required  to  do  it. 

In  June  the  results  were  sent  to  the  office  of  the 
department  for  analysis.  In  general  the  accuracy 
increased  and  the  time  necessary  to  do  the  examples 
shortened  as  the  children  worked  each  successive  form. 
In  addition  to  this,  the  data  gathered  from  this  material 
showed : 

(a.)  That  testing  of  this  kind  would  be  likely  to 
produce  results  which  would  show  the  ability  of  children 
to  add  fractions. 

(b.)  That  children  might  be  able  to  do  non-reducible 
types  but  would  have  trouble  in  doing  types  where 
reduction  was  necessary. 

(c.)     That  the  time  required  to  do  the  twelve  examples 


12 


SCHOOL  DOCUMENT  NO.    3. 


on  each  form  varied  greatly.  The  shortest  time  recorded 
by  any  individual  was  two  minutes,  the  longest  forty-one 
minutes. 

II.     THE  WORK  OF  THE  DEPARTMENT. 

Planning  the  Testing. 

a.     Selection  of  Schools. 

In  selecting  the  schools  in  which  the  tests  in  fractions 
should  be  given,  it  was  necessary  to  take  into  considera- 
tion two  conditions: 

(a.)  That  at  least  1,000  children  should  be  tested  in 
each  grade. 

(6.)  That  these  1,000  children  should  represent  the 
different  districts  of  the  city. 

In  order  to  satisfy  these  two  conditions  at  least  one 
district  was  selected  from  each  of  the  ten  groups  *  into 
which  the  city  has  been  divided  for  testing  purposes. 
The  tests  were  given  in  twelve  districts,  and  the  following 
tables  show  the  number  of  buildings,  class  rooms  and 
grade  classes  tested,  also  the  number  of  children  in  each 
grade. 

Test  in  Addition  of  Fractions,  December,  1915. 

TABLE    I. 

Number  of  districts  tested       .       .       .       ...  .12 

Number  of  buildings  tested 12 

Number  of  class  rooms  tested  .88 
Number  of  grade  classes 91 


TABLE  2. 

Gbadb. 

Number  of  Grade 
Classes. 

Number    of 
Pupils. 

VI 

31 

32 

28 

1,265 
1,243 
1,130 

VII 

VIII     

91 

3,638 

<•■  See  School  Document  No.  10,  1915,  Boston  Public  Schools. 


TEST  IN  ADDITION  OF  FRACTIONS.  13 

6.     Construction  of  Tests. 

In  constructing  the  tests  the  department  used  all  of  the 
material  furnished  them  by  the  Committee  on  Standards 
in  Arithmetic.  By  referring  to  the  explanation  of  the 
types  on  page  5,  it  will  be  seen  that  there  are  certain 
types  which  are  similar.  As  pointed  out  in  discussing 
the  results  of  the  experimental  work  carried  on  under  the 
direction  of  the  committee,  certain  errors  were  made  in 
one  type  but  not  in  others  similar  to  it.  For  example, 
a  child  might  fail  in  doing  Type  7  where  the  common 
denominator  was  one  of  the  denominators  and  after 
adding  it  was  necessary  to  reduce  the  answer  to  the 
lowest  terms,  but  have  no  trouble  in  doing  Type  5 
where  it  was  necessary  to  find  a  common  denominator 
in  the  same  way  but  the  answer  was  non-reducible. 

This  same  child  might  have  had  difficulty  in  doing 
Type  3  because,  although  it  was  not  necessary  to  find 
a  common  denominator  because  the  fractions  were 
similar,  it  was  necessary  to  reduce  the  answer  to  the 
lowest  terms.  Thus  it  was  shown  that  a  child  who  could 
do  examples  similar  to  Type  7  would  probably  be  able 
to  do  examples  similar  to  Types  3  and  5.  The  contrary, 
however,  was  not  true  for  it  was  shown  that  some 
children  who  had  no  trouble  with  Type  5  would  have 
trouble  with  Types  3  and  7.  On  the  other  hand,  owing 
to  the  fact  that  the  least  common  denominator  was 
found  in  a  different  way  in  Type  7  than  it  was  in  Type  3, 
a  child  might  be  able  to  do  Type  3  and  not  Type  7. 
It  was  thus  decided  that  examples  similar  to  Type  7 
should  compose  one  of  the  tests.  In  this  way,  examples 
from  a  Hst  representing  Types  8,  10,  13  and  14,  together 
with  those  from  Type  7,  were  decided  upon  as  five 
of  the  tests.  It  was  later  decided  to  include  examples 
similar  to  Type  3,  as  a  means  of  determining  the  zero 
abiUty  of  children  to  do  addition  of  fractions.  That  is, 
it  was  assumed  that  any  child  who  could  not  do  Type  3 
would  not  be  able  to  add  fractions  at  all  and  a  child  who 
was  just  able  to  do  Type  3  and  no  more  would  be  able 
to  handle  only  the  simplest  form  of  addition  of  fractions. 


14  SCHOOL  DOCUMENT  NO.  3. 

For  the  purpose  of  increasing  the  number  of  examples 
in  Test  5,  examples  were  selected  from  Type  11  and 
included  with  those  from  Type  13.  The  least  common 
denominator  of  these  examples  was  found  in  the  same 
way,  but  the  result  in  one  case  was  non-reducible. 
Likewise,  since  there  were  only  a  few  examples  available 
similar  to  Type  14,  it  was  necessary  to  select  examples 
from  Type  12.  The  least  common  denominator  of  each 
of  these  types  was  found  in  the  same  way,  but  in  Type 
12  it  was  only  necessary  to  reduce  the  result  to  a  mixed 
number. 

The  following,  therefore,  shows  the  type  of  example 
used  in  each  test: 

Test  1  consisted  of  examples  similar  to  Type  3. 
Test  2  consisted  of  examples  similar  to  Type  7. 
Test  3  consisted  of  examples  similar  to  Type  8. 
Test  4  consisted  of  examples  similar  to  Type  10. 
Test  5  consisted  of  examples  similar  to  Types  11  and  13. 
Test  6  consisted  of  examples  similar  to  Types  12  and  14. 

A  folder  containing  the  six  tests'was  printed  on  sheets 
8  by  10|  inches  with  the  following  directions  printed 
on  the  outside  page: 

This  folder  contains  six  tests  in  addition  of  fractions.  You 
will  be  given  two  minutes  in  each  test  to  add  as  many  of  the 
fraction  examples  as  you  can.  Use  no  other  paper.  You  are 
not  expected  to  do  them  all.  You  will  be  marked  for  both 
speed  and  accuracy,  but  it  is  more  important  to  have  your 
answers  right  than  to  try  a  great  many  examples. 

Each  test  consisted  of  twenty-four  examples.  This 
large  number  of  examples  was  given  not  with  the  idea 
that  a  child  of  any  grade  should  be  able  to  complete 
the  entire  test  in  the  time  allowed,  but  that  there 
should  be  enough  examples  to  keep  most  of  the  children 
busy  during  the  time  allotment. 

The   following   table   shows   four   examples   selected 


TEST   IN   ADDITION  OF  FRACTIONS.  15 

from  each  test.  Space  was  left  at  the  right  and  below 
each  example  for  the  child  to  do  any  figuring  which  he 
deemed  necessary. 

TABLE  3. 

Showing    Examples    Used    in    Tests    in   Addition    of    Fractions, 
December,  1915. 

Addition  of  Fractions. —  Test  1. —  Time,  2  Minutes. 
(1)      \         (2)      f^         (3)      ^         (4)      fo 

1  _L  X  1. 

«i  Ji  i£  i2 

Addition  of  Fractions. —  Test  2. —  Time,  2  Minutes. 
(1)      I  (2)      f         .(3)      I  (4)     I 

1  A  J_  JL 

'         _6  _14  _12  15 

Addition  of  Fractions. —  Test  3. —  Time,  2  Minutes. 
(1)     I  (2)     I  (3)     f  (4)     i| 

u  1  n  2 

J5  _2  _14  _3 

Addition  of  Fractions. —  Test  4. —  Time,  2  Minutes. 
(1)       \         (2)     I  (3)      I  (4)      I 

JO  _4  _7  _8 

c.  Time  Allowance. 
When  the  department  was  preparing  the  tests,  the 
question  of  how  long  a  time  should  be  allowed  for  each 
test  became  a  vital  one.  As  shown  by  the  experiment 
of  the  committee,  the  time  used  in  doing  twelve  examples 
varies  greatly.  The  department  only  desired  to  allow 
enough  time  to  show  the  ability  of  the  pupils  to  add 
fractions.    As  a  means  of  determining  this  point,  six 


16  SCHOOL  DOCUMENT  NO.  3. 

sheets  were  prepared  similar  to  those  used  in  the  test 
given  later  in  the  elementary  schools.  The  examples 
on  these  sheets  were  given  to  a  class  of  forty  students  in 
the  Normal  School,  using  a  time  allowance  of  two 
minutes  for  each  test.  This  time  allowance  was  based 
in  part  on  the  records  of  pupils  in  using  the  practice 
material.  It  was  found  that  enough  examples  were 
completed  to  give  a  measure  of  the  ability  of  the  indi- 
vidual pupil,  and  hence  this  time  was  allowed  for  each 
test. 

d.     Giving  the  Tests  and  Correcting  the  Results. 

Following  the  plan  of  giving  the  Courtis  tests,  a 
group  of  nineteen  seniors  from  the  Normal  School  were 
trained  to  give  the  tests  in  a  uniform  manner.  The 
tests  were  given  to  1,265  children  in  Grade  VI.,  1,243 
children  in  Grade  VII.,  and  1,130  children  in  Grade 
VIII.,  on  December  9,  1915.  In  addition  to  the  instruc- 
tions printed  on  the  outside  page  of  the  folder,  the 
examiners  were  instructed  that  if  they  were  asked  by 
children  or  teachers  in  regard  to  reduction  to  lowest 
terms  or  mixed  numbers,  to  say  that  they  were  to  do 
as  they  were  in  the  habit  of  doing. 

After  giving  the  tests  the  examiners  collected  the 
folders  and  brought  them  to  the  office  of  the  depart- 
ment. All  correcting  and  tabulating  of  the  results 
was  done  under  the  direction  of  the  department.  This 
method  of  handling  the  results  insures,  among  other 
things,  a  uniformity  of  correction  as  well  as  a  uniformity 
of  giving  the  test.  Any  question  respecting  the  correc- 
tion of  the  work  which  the  examiner  might  wish  to  ask 
was  answered  by  the  person  in  charge.  If  this  question 
arose  again  on  another  paper,  it  was  answered  in  the 
same  way  as  the  previous  question.  Thus  it  was  pos- 
sible to  correct  the  papers  in  a  much  more  uniform 
manner  than  under  any  other  conditions.  In  this  cor- 
rection the  main  questions  to  be  decided  were  whether 
the  pupil  had  followed  instructions  or  not.  The  follow- 
ing general  rules  were  given  the  examiners  to  govern 
them  in  the  correction  of  the  papers. 


TEST   IN  ADDITION  OF  FRACTIONS. 


17 


(a.)  All  results  which  were  not  reduced  to  lowest 
terms  or  to  mixed  numbers  were  called  wrong. 

(6.)  The  papers  on  which  children  multiplied  or 
subtracted  the  fractions  were  counted  as  I.  N.  F.  papers 
(Instructions  Not  Followed). 

(c.)  All  other  papers,  regardless  of  how  the  child  did 
the  example,  were  scored  as  right  or  wrong. 

(d.)  The  form  of  doing  the  work  did  not  count 
against  the  child  if  his  answer  was  correct. 

Analysis  of  Results. 
a.     Achievement. 

Table  4  shows  the  results  for  the  entire  number  of 
children  tested.  In  the  first  column  is  shown  the  grade, 
followed  by  a  column  showing  the  number  of  pupils 
tested  in*  each  grade.  Under  each  test  are  given  the 
speed  medians  and  the  accuracy  medians  for  each  test 
and  grade.  The  table  is  to  be  interpreted  as  follows: 
In  Grade  VI.,  1,265  pupils  were  tested.  These  pupils 
attained  a  speed  median  of  10.7  examples  with  an  accu- 
racy median  of  79.6  per  cent  in  Test  1.  In  Test  2  the 
speed  median  is  less,  falUng  to  7.7  examples  and  the 
accuracy  median  falling  to  65.6  per  cent.  Thus,  read- 
ing across  the  page  on  the  first  line  will  be  found  the 
two  medians  for  Grade  VI.  The  table  shows  the  same 
facts  for  Grades  VII.  and  VIII. ,  respectively. 


TABLE  4. 
Summary  Sheet  —  City  Medians. 

Addition  of  Fraclions,  December,  1915. 


fl 

TkstI. 

Tb8t2. 

Ti»t3. 

Test  4. 

Tb8t5. 

Test  6. 

Grade. 

is 

a 

.2 
-o'S 

P,S 

a 

i 

is 

d 

0^ 

si 

is 

i 
■A 

II 

CL. 

CO 

< 

m 

< 

m 

<; 

no 

< 

m 

< 

OQ 

< 

VI 

1.265 

10.7 

79.6 

7.7 

65.6 

5.5 

41.9 

4.0 

69.5 

4.6 

51.0 

4.4 

48.6 

vn 

1,243 

16.5 

86.6 

10.1 

1 

72.9 

7.3 

46.1 

5.3 

69.2 

6.3 

54.9 

5.7 

48.1 

vni 

1.130 

20.7 

88.2 

1  11.6 

74.4 

8.4 

47.4 

6.0 

67.8 

6.9 

52.4 

6.4 

46.5 

18  SCHOOL  DOCUMENT  NO.  3. 

How  far  these  figures  may  be  relied  upon  as  setting  a 
standard  to  be  attained  in  addition  of  fractions  is 
doubtful.  Many  children  paid  no  attention  to  reduc- 
tion to  mixed  numbers  or  reduction  to  lowest  terms; 
thus  they  did  an  abnormally  large  number  of  examples 
with  a  large  number  incorrect.  This  makes  the  speed 
median  high  and  the  accuracy  median  low.  The  only 
thing  needed  to  make  correct  many  of  these  incorrect 
results  was  the  reduction  to  lowest  terms.  If  this  had 
been  done  as  the  child  proceeded  with  his  work,  he 
would  necessarily  have  done  fewer  examples  with 
greater  accuracy. 

In  the  first  test  there  were  many  cases  of  children  who 
wrote  answers  to  the  entire  twenty-four  examples  and 
reduced  the  results  to  lowest  terms  until  the  time  expired. 
The  result  was  a  child  might  have  had  twenty-four 
examples  attempted  with  five  examples  right.  If 
instead  of  doing  his  work  in  this  way  he  had  reduced 
each  example  to  lowest  terms  as  he  proceeded,  his 
number  attempted  would  have  been  less  and  his  accuracy 
greater.  This  happened  in  so  many  cases  in  all  of  the 
tests  that  the  standard  medians  in  speed  should  probably 
be  lower  than  those  represented  in  the  table  and  the 
standard  accuracy  higher.  However,  whether  the 
results  shown  in  Table  4  are  higher  or  lower  than  a  stand- 
ard ought  to  be,  they  show  what  the  department  aimed 
to  find  out,  namely,  what  the  facts  are  concerning  the 
abiUty  of  elementary  school  children  to  add  fractions. 

b.  Kinds  of  Errors. 
In  constructing  the  tests  it  was  intended  to  arrange 
them  in  the  order  of  difiiculty.  If  we  judge  whether 
this  has  been  done  or  not  upon  the  median  speed  with 
which  a  given  grade  performed  the  examples  of  a  given 
test.  Table  4  shows  that  apparently  Test  4  is  more 
difficult  than  either  Test  5  or  6  because  the  number  of 
examples  done  in  the  time  allowed  is  slightly  less  in  all 
three  grades  than  was  accomplished  in  the  two  following 
tests.     The  accuracy  with  which  Test  4  was  done,  how- 


TEST   IN  ADDITION   OF  FRACTIONS.  19 

ever,  is  slightly  greater  than  in  either  of  the  two  following 
tests.  This  result  may  be  due  in  part  to  the  fact  that 
the  examples  represented  by  Type  4  are  much  more 
numerous  than  in  either  of  the  other  types  represented 
in  the  test.  Therefore,  chance  would  make  it  probable 
that  the  child  would  deal  with  this  type  of  examples 
much  oftener  than  with  any  of  the  remaining  types. 
This  would  tend  toward  greater  accuracy.  The  common 
denominator  in  this  particular  type  may  be  larger  and 
therefore  might  have  a  tendency  to  cause  slower  motor 
reactions,  thus  causing  a  slight  reduction  in  the  number 
of  examples  attempted.  However,  in  view  of  the 
results  indicated,  it  would  seem  that  Tests  1  and  2  were 
easier  than  any  of  the  others  and  that  Tests  3,  4,  5,  6 
are  relatively  of  nearly  the  same  difficulty.  This  is  also 
borne  out  by  the  number  of  failures  in  these  four  tests. 

In  studying  the  Tesults  of  Test  3  it  will  be  noticed 
that  although  the  speed  median  is  only  slightly  smaller 
than  in  Test  2  and  larger  than  in  Test  4  or  5,  the  accuracy 
median  is  very  low.  This  fact  was  so  noticeable  that  it 
led  to  a  detailed  study  of  this  test  in  order  to  discover 
the  reason  for  this  low  percentage  of  accuracy.  In  order 
to  obtain  a  correct  answer  in  the  examples  in  this  test  it 
was  necessary  for  the  children,  after  reducing  to  a 
common  denominator  and  adding  the  numerators,  to 
reduce  the  result  thus  obtained  to  both  lowest  terms 
and  a  mixed  number.  Under  these  conditions,  the  low 
accuracy  was  found  to  be  due  to  three  causes: 

(a.)  19.1  per  cent  paid  no  attention  to  reducing  to 
mixed  numbers. 

(6.)  17.6  per  cent  paid  no  attention  to  the  reduction 
to  lowest  terms. 

Nearly  half  of  these  paid  no  attention  to  either 
reduction. 

(c.)  Although  they  reduced  to  lowest  terms  and  to  a 
mixed  number,  4.5  per  cent  expressed  the  answer  in  such 
a  way  that  it  did  not  tell  the  truth.  For  example,  in 
adding  f  and  H  they  obtained  H  =  lA  =  i  instead  of 


20 


SCHOOL  DOCUMENT  NO.    3. 


In  some  cases  where  this  was  done,  not  only  in  Test  3 
but  in  Test  6,  it  was  questionable  whether  the  children 
really  understood  that  the  answer  was  1|.  Many  of 
them  did  the  example  in  the  space  at  the  right  provided 
for  that  purpose  and  finally  wrote  their  answer  under 
the  Une  beneath  the  example  and  wrote  only  the  fraction, 
leaving  out  the  whole  number  entirely.  This  would  be 
of  vital  importance  if  mixed  numbers  were  being  added. 
Further,  it  seems  that  if  we  can  tell  the  truth  in  our 
arithmetic  work  we  should  by  all  means  do  so.  Probably 
no  sign  is  more  abused  and  misused  than  the  sign  of 
equaUty  and  if  we  could  reinforce  the  meaning  of  the 
word  "equals"  in  this  particular  case,  we  should  take 
advantage  of  the  opportunity  to  make  the  expression 
which  is  written  tell  the  truth. 


TABLE  5. 
Per  Cent  of  Pupils  Obtaining  Zero  Examples  Right  in  Fraction  Test. 


VIII. 

VII. 

VI. 

Number  of  pupils 

1,130 
20.6 
35.0 
44.3 
43.3 
30.4 
42.9 

1,243 
22.0 
24.4 
46.2 
28.6 
28.2 
33.8 

1,265 

Test  1 

27.0 

Test  2 

34.2 

Test  3 

46.6 

Test  4 

31.7 

Test  5 

34.5 

Test  6     

39.8 

One  method  of  determining  the  standing  of  a  class  is 
to  show  the  percentage  of  children  who  obtain  a  score  of 
zero  in  the  examples  right.  Table  5  shows  the  per  cent 
of  pupils  in  each  grade  who  had  no  examples  right  in 
each  separate  test.  That  is,  20.6  per  cent  of  the  children 
in  Grade  VIII.  had  no  examples  right  in  the  simplest 
test.  Test  1;  22  per  cent  of  the  seventh  grade  and  27 
per  cent  of  the  sixth  grade  did  no  better  in  the  same 
test.  The  large  per  cent  of  the  pupils  obtaining  a  zero 
score  in  the  third  test  is  due,  as  pointed  out  in  the 
foregoing,  to  the  fact  that  the;y  did  not  reduce  either  to 


TEST  IN   ADDITION   OF  FRACTIONS.  21 

lowest  terms  or  to  mixed  numbers.  The  failure  to 
perform  the  necessary  reductions  is  the  cause  of  the 
high  percentage  of  zero  scores  in  all  grades  and  in  all 
tests.  The  relatively  low  per  cent  in  Test  5  is  probably 
due  to  the  construction  of  the  test.  As  previously 
indicated,  the  test  consists  of  examples  from  Types  11 
and  13.  The  inclusion  of  Type  11  was  found  to  be  a 
help  to  the  children  because  many  who  were  unable  to 
obtain  a  correct  answer  to  the  first  example  which  was 
taken  from  Type  13,  because  they  did  not  reduce  to  the 
lowest  terms,  were  able  to  obtain  a  correct  answer  to 
the  second  example  which  was  taken  from  Type  1 1  and 
therefore  did  not  have  to  be  reduced. 

The  large  per  cent  of  zero  scores  in  Test  6  may  be  due 
partially  to  the  method  of  finding  the  least  common 
denominator.  The  question  of  factoring  has  not  been 
emphasized  very  largely  in  the  last  few  years,  and  may 
raise  a  question  as  to  whether  such  examples  should 
have  been  included  in  the  tests.  In  making  up  the  test, 
however,  fractions  with  large  denominators  were  care- 
fully avoided.  In  fact,  in  no  case  was  the  denominator 
of  either  fraction  larger  than  12  and  the  least  common 
denominator  in  no  case  exceeded  60.  It  is  the  opinion 
of  the  department  that  such  fractions  are  large  enough 
for  ordinary  purposes,  but  that  in  no  case  should  the 
denominator  of  the  fractions  to  be  added  exceed  16 
and  that  the  least  common  denominator  should  be  less 
than  100.  This  should  make  it  possible  for  most  children 
to  determine  the  least  common  denominator  by  inspec- 
tion and  make  it  unnecessary  to  teach  the  forms  of 
finding  the  least  common  denominator  by  factoring  as 
taught  in  the  various  arithmetics. 

c.  Proportion  of  Failures. 
There  are  two  classes  of  children  who  obtain  a  score  of 
zero  in  their  work.  To  the  first  class  belong  the  children 
who  get  zero  because  of  inaccurate  work.  Their  method 
of  doing  the  example  is  correct.  The  second  class  include 
those  children  who  get  zero  because  they  do  not  know 
the  method  to  be  followed. 


22 


SCHOOL  DOCUMENT  NO.   3. 


In  undertaking  to  find  how  many  made  a  failure  in 
adding  fractions,  only  the  second  class  were  taken  into 
consideration.  That  is,  it  was  counted  a  failure  on  the 
part  of  the  child  only  when  he  showed  by  his  work  that 
he  had  no  conception  as  to  the  meaning  of  addition  of 
fractions.  The  number  of  examples  which  he  performed 
was  generally  small  and  the  work  accompanying  those 
examples  showed  that  he  had  little  or  no  idea  of  reduction 
to  a  common  denominator.  In  many  cases  he  undertook 
.  to  add  the  fractions  without  finding  a  common  denomina- 
tor at  all.     These  failures  fell  under  three  heads. 

(a.)  Adding  of  numerators  without  considering  a 
common  denominator.  In  order  to  make  the  answer  a 
fraction,  the  sums  may  have  been  placed  over  the  sum 
of  the  denominators,  the  product  of  the  denominators, 
or  over  either  one  of  the  denominators. 

(6.)  Adding  all  the  figures  as  integers  and  either 
placing  the  sum  over  some  denominator  or  allowing  it 
to  stand  as  an  integer. 

(c.)  Apparently  impossible  ways  of  obtaining  an 
answer. 

Table  6  shows  there  were  70  children  or  6.2  per  cent 
in  Grade  VIII. ,  144  children  or  11.6  per  cent  in  Grade 
VII.  and  108  children  or  8.5  per  cent  in  Grade  VI.,  who 
made  such  failures  in  Test  1.  In  Test  2,  182  or  16.1  per 
cent  in  Grade  VIIL,  218  or  17.5  per  cent  in  Grade  VII., 
and  211  or  16.7  per  cent  in  Grade  VI.  failed,  and  so  on. 


TABLE  6. 
Number  of  Pupils  and   Per  Cent   Making  Absolute   Failures  in 
Each  Test  in  Addition  of  Fractions. 


VIII. 

VII. 

VI. 

Pupils. 

Per  Cent. 

Pupils. 

Per  Cent. 

Pupils. 

Per  Cent. 

Teet  1 

70 
182 
171 
196 
193 
193 

6.2 
16.1 
16.1 
17.4 
17.1 
17.1 

144 
218 
249 
261 
256 
256 

11.6 
17.5 
20.0 
20.9 
20.6 
20.6 

108 
211 
237 
279 
253 
240 

8.5 

Test  2 

16.7 

Test  3      

18.7 

Teat  4 

22.1 

Test  5 

20.0 

Test  6 

18.9 

TEST  IN   ADDITION  OF  FRACTIONS. 


23 


Notice  that  about  17  per  cent  in  Grade  VIII.,  20  per 
cent  in  Grade  VII.  and  20  per  cent  in  Grade  VI.  failed 
to  do  Tests  3,  4,  5  and  6.  This  bears  out  what  has 
been  aheady  stated,  namely,  that  these  four  tests  are 
apparently  of  nearly  the  same  difficulty. 

TABLE  7. 

Number  of  Pupils  and  Per  Cent  Making  Absolute  Failures    in 

Combined  Tests  in  Addition  of  Fractions. 


VIII. 

VII. 

VI. 

Pupils. 

Per  Cent. 

Pupils. 

Per  Cent. 

Pupils. 

Per  Cent. 

Testa  1-2-3-4-5-6 

Tests  2-3-4-5-6 

Teats  3-4-5-6 

57 

96 

7 

20 

7 

6 

5.0 

8.5 

.6 

1.8 

.6 

.5 

144 

71 

28 

12 

1 

0 

11.6 

5.7 

2.3 

1.0 

.1 

.0 

104 

97 

16 

28 

1 

2 

8.2 
7.7 
1.3 

Tests  4-5-6 

2.2 

Tests  5-6    

.1 

Test  6". 

.2 

Table  7  shows  the  number  of  pupils  and  the  per 
cent  making  absolute  failures  in  the  combined  tests  in 
addition  of  fractions.  That  is,  57  pupils  or  5  per  cent 
of  Grade  VIII.,  144  pupils  or  11.6  per  cent  of  Grade 
VII.,  104  pupils  or  8.2  per  cent  of  Grade  VI.  failed  in 
solving  a  single  example  in  any  test.  If,  as  pointed 
out  earUer  in  this  report.  Test  1  shows  the  zero  ability 
of  pupils  to  do  addition  of  fractions,  then  from  5  per 
cent  to  11  per  cent  possess  this  zero  abiUty.  Also  96 
pupils  or  8.5  per  cent  of  Grade  VIII.,  71  pupils  or  5.7 
per  cent  of  Grade  VII.,  97  pupils  or  7.7  per  cent  of 
Grade  VI.  were  unable  to  do  anything  beyond  the  first 
test.  That  is,  from  5  per  cent  to  8  per  cent  are  just 
able  to  do  a  simple  example  in  addition  of  fractions 
when  there  are  two  addends.  The  table  shows  that, 
on  the  whole,  if  a  child  could  do  the  first  two  tests,  he 
could  do  the  following  four  tests.  Only  a  very  small 
per  cent  failed  in  these  lafet  tests. 

d.     Causes  of  Errors. 
In  general  there  are  three  causes  of  inaccuracy, 
(a.)     Fundamental  faults;   that  is,  the  children  have 


24  SCHOOL  DOCUMENT  NO.  3. 

little  or  no  idea  how  to  find  the  least  common  denomina- 
tor, how  to  reduce  the  fractions  to  this  least  common 
denominator,  how  to  reduce  to  lowest  terms  or  to 
mixed  numbers.  Many  such  childreii  make  a  proper 
fraction  equal  to  a  whole  or  mixed  number :  for  example, 

3    92 

(6.)  Faults  that  are  easily  corrected.  For  example, 
the  reduction  to  lowest  terms  and  to  mixed  numbers 
in  most  cases  requires  only  an  explanation  by  the 
teacher,  and  then  an  insistence  that  all  papers  passed 
in  shall  have  the  result  in  its  lowest  terms  or  shall  be 
reduced  to  a  whole  or  mixed  number. 

(c.)  Inaccuracy  in  the  work  which  the  child  does. 
These  are  the  same  inaccuracies  that  occur  in  the 
manipulation  of  whole  numbers  and  in  all  cases  tend 
toward  low  scores  in  speed  or  in  accuracy. 

e.     Methods  Used  in  Doing  the  Examples. 

Some  of  the  methods  used  in  doing  the  examples 
were  ineffective.  In  some  cases  they  caused  the  pupil 
to  waste  the  time  which  he  had  at  his  disposal;  in  other 
cases  the  methods  trained  him  to  think  in  a  wasteful 
manner  or  not  to  think  at  all. 

Approximately  one  third  found  it  necessary  to  reduce 
the  fractions  to  a  common  denominator  in  the  first 
test  when  the  fractions  were  already  similar.  Some  of 
these  children  wrote  the  fractions  over  a  common 
denominator,  using  for  a  common  denominator  the 
denominator  of  the  similar  fractions.  Others,  not 
noticing  that  the  fractions  already  had  a  common 
denominator,  used  some  multiple,  making  the  necessary 
reductions.  For  example,  many  children  added  -^  and 
T^  by  reducing  the  fractions  to  a  common  denominator 
of  196.  In  many  cases  they  then  made  errors  in  their 
work,  thus  obtaining  an  incorrect  answer  to  the  example. 
Even  if  carried  through  correctly,  this  is  an  ineffective 
and  wasteful  way  of  doing  such  examples. 

Another  method  used  by  many  individuals  consisted 
of    finding    the    least    common    denominator    of    such 


TEST  IN   ADDITION  OF   FRACTIONS.  25 

fractions  as  |  and  ^\  by  finding  the  least  common  mul- 
tiple of  the  denominators  by  short  division  as  taught 
in  many  of  the  arithmetics.  In  such  cases  the  following 
was  found: 

2)8  —  16 
2)4—   8 

2)2—   4  2x2x2x2=  16 

2)1-    2 
1—    1 

This  indicates  that  the  child  is  doing  no  thinking. 
The  process  is  a  wholly  mechanical  one  and  is  not  devel- 
oping in  the  child  the  habit  of  correct  or  independent 
work.  Such  a  method  gives  the  child  no  basis  for 
judging  whether  the  answer  is  reasonable  or  not.  No 
child  who  is  thus  tied  to  the  mechanics  of  the  operation 
has  any  opportunity  for  thinking  about  the  result  he  is 
after.  Oral  work  should  take  care  of  finding  the  common 
denominator  in  all  cases  of  fractions  with  such  small 
denominators. 

/.     Effect  of  Practice. 

The  tests  in  addition  of  fractions  were  given  to  three 
of  the  schools  in  which  the  material  prepared  by  the 
Committee  on  Standards  in  Arithmetic  was  used.  For 
piu'poses  of  comparison  the  department  selected  the 
grade  in  which  the  children  who  had  used  the  practice 
material  were  enrolled  in  their  respective  schools.  The 
papers  of  the  selected  grades  in  the  three  schools  (termed 
in  the  following  table,  schools  A,  B,  C)  were  then 
sorted  and  the  results  tabulated  separately.  Follow- 
ing this,  the  papers  of  the  children  who  used  the  practice 
material  (termed  in  the  table  the  ''selected  group") 
were  then  sorted  and  the  median  scores  computed. 


26 


SCHOOL  DOCUMENT  NO.  3. 


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TEST  IN  ADDITION   OF  FRACTIONS.  27 

Table  8  shows  the  median  scores  of  these  three  groups 
of  children.  It  is  to  be  interpreted  as  follows:  In  the 
line  following  the  word  "city"  is  shown  the  speed  median 
and  accuracy  median  for  each  test  for  Grade  VI.  in  all 
the  schools  tested,  as  already  shown  in  Table  4.  In 
the  next  line  the  table  shows  that  school  A  had  a  speed 
median  of  12.9  examples  with  an  accuracy  median  of 
48.8  per  cent  in  Test  1,  and  a  speed  median  of  9  examples 
with  an  accuracy  median  of  43.6  per  cent  in  Test  2,  and 
so  on.  The  selected  group  of  school  A  has  a  speed 
median  of  15  examples  with  an  accuracy  median  of  87.5 
per  cent  in  Test  1,  and  so  on. 

The  selected  group  in  school  A  shows  an  advantage 
in  speed  in  each  test  over  the  city-wide  results.  There 
is  not  always  a  superiority  in  this  group  over  the  results 
of  the  grade  distribution  of  its  own  school.  In  spite 
of  obtaining  a  lower  median  in  speed,  however,  the 
group  does  surpass  its  own  school  in  each  test  in  accuracy. 
It  fails  to  pass  the  city-wide  accuracy  median  in  Tests 
3,  4  and  6.  The  selected  group  of  school  C  surpasses 
the  city-wide  results  in  speed  in  each  test  and  its  own 
school  in  all  but  Tests  4  and  5.  This  group  also  sur- 
passes both  city-wide  results  and  the  results  of  its  own 
school  in  accuracy. 

The  median  for  school  B  was  comparatively  high  and 
the  superiority  of  the  selected  group  does  not  show  so 
strikingly.  It  is  plain,  however,  that  even  the  small 
amount  of  individual  training  which  was  given  during 
the  spring,  and  may  have  been  continued  during  the 
fall,  has  produced  favorable  results. 

SUMMARY  AND   CONCLUSIONS. 

1.  The  factors  that  enter  into  the  problem  of  adding 
fractions  are  much  more  complex  than  those  that  enter 
into  the  problem  of  adding  integers. 

2.  The  errors  were  largely  due  to  failing  of  pupils 
to  reduce  consistently  either  to  lowest  terms  or  to  mixed 
numbers. 


This  failing  on  the  part  of  many  children  to  use  the 
principle  of  reduction  would  seem  to  indicate  that  the 
method,  now  largely  in  use,  of  teaching  such  reductions 
|by  themselves,  has  failed  to  produce  satisfactory  results. 
In  view  of  this  fact,  would  it  not  be  well  to  teach  reduc- 
tions, as  such,  in  connection  with  the  subject  of  addition 
of  fractions?  This  would  at  least  make  a  closer  con- 
nection between  the  two  operations,  and  thereby  tend 
to  form  the  habit  of  writing  the  answer  in  its  best  form. 

3.  Eight  per  cent  in  Grade  VI.,  11  per  cent  in  Grade 
VII.  and  5  per  cent  in  Grade  VIII.  were  unable  to  do  the 
simplest  problems  in  the  addition  of  fractions. 

4.  Drilling  and  individual  work  given  children  in  the 
schools  in  the  spring  showed  its  efifect  in  the  late  fall. 
This  was  evidenced  by  an  increase  in  both  speed  and 
accuracy  over  that  obtained  in  the  entire  city  and  in  two 
cases  over  that  shown  by  the  whole  number  of  pupils 
in  the  grade  in  which  the  selected  groups  were  enrolled. 


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